These aren't really great "moments" since some of the developments I describe evolved over decades. So I'll claim 'literary license' (which means I like the way the phrase sounds and don't give a hoot what the words actually mean ....)
I'll try to add as many of these as I can. As I've said before, success in mathematics requires a 'mental syntax' or 'mindset' to be developed. These essays describe some of the great questions of mathematics, and reading how solutions to these problems evolved is instructive. Besides - it's fun. Believe me :-)
Archimedes and Approximations of pi.pdf - Look what it did to the name (it's supposed to be approximations of pi ....). Archimedes may have been the greatest mathematician of all time, and this is instructive. I've included it here because you can see how the "ancients" treated our modern idea of limit intuitively, and in a way that avoided grappling with the idea of infinity. Getting your hands wet with Geometry will benefit you on the AP Exam (metaphor sucks, I know - go tell Foltz)
Newton and the Binomial Theorem.pdf - Newton was an incredible genius and I didn't really appreciate that very much until this past summer (2008). Here is one of his greatest achievements that has great practical value (if your calculator batteries die!) and also illustrates that some functions can be expressed as 'infinite term "polynomials"'. This will become valuable later (if not in this course, then for later courses). And it's STILL fun!
Cantor - The 19th Century Tackles Infinity.pdf - I'm not allowed to medicate students, or my reward for reading this entire essay would be a handful of aspirin. Everyone should look at this once in their lifetime. Very little relevance to the AP exam!
Archimedes and Squaring the Parabola.pdf - Not really Archimedes' approach, but an activity that helps you understand one of the types of great questions asked throughout Western civilization. It turns out Archimedes anticipated Newton's and Leibniz's work in discovering the calculus by about 2000 years. An extraordinary achievement.
I'll try to add as many of these as I can. As I've said before, success in mathematics requires a 'mental syntax' or 'mindset' to be developed. These essays describe some of the great questions of mathematics, and reading how solutions to these problems evolved is instructive. Besides - it's fun. Believe me :-)
Archimedes and Approximations of pi.pdf - Look what it did to the name (it's supposed to be approximations of pi ....). Archimedes may have been the greatest mathematician of all time, and this is instructive. I've included it here because you can see how the "ancients" treated our modern idea of limit intuitively, and in a way that avoided grappling with the idea of infinity. Getting your hands wet with Geometry will benefit you on the AP Exam (metaphor sucks, I know - go tell Foltz)
Newton and the Binomial Theorem.pdf - Newton was an incredible genius and I didn't really appreciate that very much until this past summer (2008). Here is one of his greatest achievements that has great practical value (if your calculator batteries die!) and also illustrates that some functions can be expressed as 'infinite term "polynomials"'. This will become valuable later (if not in this course, then for later courses). And it's STILL fun!
Cantor - The 19th Century Tackles Infinity.pdf - I'm not allowed to medicate students, or my reward for reading this entire essay would be a handful of aspirin. Everyone should look at this once in their lifetime. Very little relevance to the AP exam!
Archimedes and Squaring the Parabola.pdf - Not really Archimedes' approach, but an activity that helps you understand one of the types of great questions asked throughout Western civilization. It turns out Archimedes anticipated Newton's and Leibniz's work in discovering the calculus by about 2000 years. An extraordinary achievement.